The current president has repeatedly criticized the agreement as one of the worst deals for the USA that didn't stop Iran's work toward a nuclear weapon.
"I am asking for WORLD PEACE, nothing less!"
The first of two rounds of USA sanctions kicked in at 12:01 am (0431 GMT), targeting Iran's access to United States banknotes and key industries including cars and carpets.
Trump declared the landmark 2015 agreement had been "horrible", leaving the Iranian government flush with cash to fuel conflict in the Middle East.
Some re-imposed sanctions took effect overnight and tougher ones relating to oil exports will begin in November.
Rouhani said Monday, "I don't have preconditions".
The United States has reimposed economic sanctions on Iran that were originally put in place to pressure the country to limit its nuclear program and later lifted under a 2015 global agreement.
Most foreign businesses will not invest in Iran and will pull out, opting instead to stay in the good graces of the U.S.
With US sanctions set to return against Iran on Tuesday, Zarif acknowledged there were hard times ahead.
A lot of the practical effects of the sanctions are yet to be seen, as they depend heavily on which nations go along with U.S. demands. Some of the U.S.'s closest allies have signaled they wish to continue trade with Iran, especially when it comes to oil.
"We are determined to protect European economic operators engaged in legitimate business with Iran", she said in a statement.
Iranian Foreign Minister Mohammad Javad Zarif told reporters the global reaction to Trump's move showed that the United States was diplomatically "isolated".
Many large European firms are leaving Iran for fear of U.S. penalties, and Mr Trump warned of "severe consequences" against firms and individuals that continued to do business with Iran. It has added to tensions inside Iran, which has seen days of protests and strikes in multiple towns and cities over water shortages, high prices and wider anger at the political system.
Iranian authorities recently also arrested 45 people, including the Central Bank's deputy chief, as part of a crackdown on financial fraud, according to judiciary spokesman Gholamhossein Mohseni Ejehi.
"Since the deal was reached, Iran's aggression has only increased", Trump said.
The European commission today put in place never-before-used legislation to protects the continent's firms from United States sanctions which not only target Tehran but their trading partners.
There is a new confidence in Washington's foreign policy.
Israeli Prime Minister Benjamin Netanyahu hailed the renewed sanctions as "an important moment for Israel, for the U.S., for the region, for the whole world". But most Iranians have yet to see major economic improvement, and the prospect that Washington would re-impose sanctions helped drive a collapse in the value of Iran's currency this year, raising the cost of imports.